My math is rusty so need help on integrated exponential decay (total from t=0 to T)

Billy T

Billy T

Use Sugar Cane Alcohol car Fuel
Valued Senior Member
I want to know the CO2 equivalence, CO2e, of Methane for some years not given in the literature and know the atmospheric half life of a CH4 molecule is now* 12.4 years, and know that each molecule of CH4's CO2e(20) is 86 for a 20 year period and its CO2e(100) = 34 for 100 years. (From table on last page here: http://en.wikipedia.org/wiki/Global-warming_potential)

I will assumed an exponential decay* of N atoms of CH4 released at t = 0. I.e. N(t) = N{(0.5)^(t/12.4)} where t is in years. I also assume that the CO2e in / for Y years is proportional to the average number of CH4 molecule existing during those Y years.

I.e. that that data is telling me, I think, that the integral from 0 to 20 of N(t) dt with that result divided by 20 to get an effective constant number of CH4 molecules, I'll call N(20)a, is proportional to 86
AND
that the integral from 0 to 100 of N(t) dt with that result divided by 100 to get an effective constant number of CH4 molecules, N(100)a. is proportional to 34.

I, for example, want to know the CO2e of methane over the next 5 and 10 years. I. e. given the 20 & 100 year CO2e results and my assumption they scale with the average number of molecules existing during the period of interest, it should be possible to compute N(Y)a for any Y year period.

I think perhaps only one of the link's results, say 86, may permit this a calculation. In which case, the other (34) can be a test of idea than the COe for CH4 is directly proportional to N(20)a and see if it is (or is not) 34.


* Actually the half life is slowly increasing (as CH4 is mainly destroyed in reaction with the OH- radical, but both are destroyed in that reaction, and current release rate of CH4 is faster than the OH- radical production rate. About a decade ago, when there higher OH- concentration, the CH4 half life was slightly less than 10 years, not the current 12.4) Thus, the rate of "exponential decay" slowing, but it is conservative to neglect this.
 
$$\Huge N(t) = N_0 2^{ - \frac{t}{t_{\frac{1}{2}}}} \quad \Rightarrow \quad \frac {1}{T} \int_0^{T} N(t) dt = \frac{N_0 \, t_{\frac{1}{2} }}{ ( \ln 2 )\, T} \left( 1 - 2^{-\frac{T}{t_{\frac{1}{2}}} \right)$$

But the web page doesn't mention half-life ($$t_{\frac{1}{2}}$$) but instead, lifetime ($$\tau$$)

Thus comparing:
$$T_a = 5 T_b$$ versus $$T_b = 20$$ when $$\tau_{CH_4} = 12.6$$ we have the following relations:
$$ \frac{ K \tau_{CH_4} \left( 1 - e^{-\frac{T_a}{\tau_{CH_4}}} \right) }{ \tau_{CO_2} \left( 1 - e^{-\frac{T_a}{\tau_{CO_2}}} \right) } = 34 \\ \frac{ K \tau_{CH_4} \left( 1 - e^{-\frac{T_b}{\tau_{CH_4}}} \right) }{ \tau_{CO_2} \left( 1 - e^{-\frac{T_b}{\tau_{CO_2}}} \right) } = 86$$
Or substituting $$T_a = 5 T_b, x = e^{-\frac{T_b}{\tau_{CO_2}}$$ we get:
$$ \frac{ K \frac{126}{10} \left( 1 - e^{-\frac{1000}{126}} \right) }{ \tau_{CO_2} \left( 1 - x^5 \right) } = 34 \\ \frac{ K \frac{126}{10} \left( 1 - e^{-\frac{200}{126}} \right) }{ \tau_{CO_2} \left( 1 - x \right) } = 86$$

Dividing the bottom equation by the top equation we get:
$$\frac{ 1 - e^{-\frac{200}{126}} }{ 1 - e^{-\frac{1000}{126}} } \frac{ 1 - x^5 }{1 -x } = \frac{86}{34}$$
or
$$x^4+x^3+x^2+x + 1 - \frac{43 \left(1-e^{-\frac{500}{63}} \right) }{17 \left(1-e^{-\frac{100}{63}} \right) } = 0$$ which has just 1 positive root near x = 0.7712118781579813176561830585466523934427
which requires
$$\tau_{CO_2} = -\frac{20}{\ln ( 0.771212 ) } = 76.98$$
and $$ K = 151 $$

Thus (if this reasoning and model are correct) $$\textrm{GWP}_{CH_4}(T) = 151 \times \frac{ 12.6 \left( 1 - e^{-\frac{T}{12.6}} \right) }{ 77 \left( 1 - e^{-\frac{T}{77}} \right) }$$
which decreases monotonically from 151 at T=0 to an asymptote for large T near 24.7
 
Many, many thanks. You have researched the problem better than I have and done it correctly; however, my real interest was not the equlivalent CO2e equal to an impulse release of CH4, the so called GWP, but to get the relative damage or average radiative forcing for periods shorter than the 20 years I could find in the literature.

My method for doing this basically ignored the removal of CO2 as I knew it was very complex with some remaining in the atmosphere for 1000+ years (as your 2nd link's abstract below at end shows). Also adding to the complexity is that in the first week CO2 molecule 1478 of 10,000 released in the impulse may become part of some growing grass, and then in the fall as the grass decays, return to the air to again help keep earth warmer. Etc. for other temporary CO2 sinks.

My reasoning was that the damage done by 10,000 CH4 molecules released in an impulse at t=0 in 5 years with basically an exponential decaying existence (as number removed each minute was proportional to how many existed at the start of the minute) would do the same damage as a the average number existing during those 5 years with none being removed. I. e. I needed help calculating the N(5)a when N(0) = 10,000 when the GWP of N(20) = 86 and N(100) = 34 were known GWPs. I think that means I would need to set N(0) to some value (not 10,000) and apply my method (not the fully correct one) to reproduce the 86 value, with t= 20, and see what that N(0) gave for the N(100)'s GWP value instead of 34, as and accuracy test (if N(0) can be found from requirement that N(20) = 86 alone, as I expect some one clever like you can do.)

Do you agree that this non-decreasing set of N(5)a molecules will have essentially the same effect on climate during 5 years as the true, declining set of N(t) molecules does for 5 years? If so, I would still like your (or anyone's) help in evaluating N(t)a. for t < 20. Again Many, many thanks not only for your effort, but also for that very informative first link, which will take me some time to fully benefit from it.

BTW, I'm almost sure your K is what I call N(0) the number needed to make the GWP of N(5) = 77 (a more correct GWP than 86 for the 20 year period?). Am I correct on this?

For what I am interested in, the fact that the GWP does not fall to zero but is 24.7 after 1000+ years is of no importance (nor is the IPCC's 100 year value, as I'm worried about mankind being able to survive* for next five or so decades.) because there is essntailly no damage being done by the impules of CH4 releasted at t =0 at t > 100 as that is approximately 8 half lives with less than 0.004 of the impulse released molecules still remaining undestroyed.
---------
* Both average humidity and temperature are increasing, and probably both with accelerated rate of increase. A wet-bulb temperature of your environment of only 35C is quickly fatal as you can not dump the ~100W your 37C body is generating even just sitting in a chair.
You have been active in the "Climate-gate" thread so probably have read: http://www.sciforums.com/showthread.php?97892-Climate-gate&p=3186464&viewfull=1#post3186464 and many of my later posts in that thread, but I give the link for others who may not realize the risk man is running with such rapid burning of fossil fuels.

The reason I am interested in the shorter period GWP of CH4 is that I think it can be removed from the air profitably (by claiming the Carbon Credits for doing so) - How I will post soon in the Climate Gate thread.

--------------------------------------------
Abstract. The responses of carbon dioxide (CO2) and other climate variables to an emission pulse of CO2 into the atmosphere are often used to compute the Global Warming Potential (GWP) and Global Temperature change Potential (GTP), to characterize the response timescales of Earth System models, and to build reduced-form models. In this carbon cycle-climate model intercomparison project, which spans the full model hierarchy, we quantify responses to emission pulses of different magnitudes injected under different conditions. The CO2 response shows the known rapid decline in the first few decades followed by a millennium-scale tail. For a 100 Gt-C emission pulse added to a constant CO2 concentration of 389 ppm, 25 ± 9% is still found in the atmosphere after 1000 yr; the ocean has absorbed 59 ± 12% and the land the remainder (16 ± 14%). The response in global mean surface air temperature is an increase by 0.20 ± 0.12 °C within the first twenty years; thereafter and until year 1000, temperature decreases only slightly, whereas ocean heat content and sea level continue to rise. Our best estimate for the Absolute Global Warming Potential, given by the time-integrated response in CO2 at year 100 multiplied by its radiative efficiency, is 92.5 × 10−15 yr W m−2 per kg-CO2. This value very likely (5 to 95% confidence) lies within the range of (68 to 117) × 10−15 yr W m−2 per kg-CO2.

Estimates for time-integrated response in CO2 published in the IPCC First, Second, and Fourth Assessment and our multi-model best estimate all agree within 15% during the first 100 yr. The integrated CO2 response, normalized by the pulse size, is lower for pre-industrial conditions, compared to present day, and lower for smaller pulses than larger pulses. In contrast, the response in temperature, sea level and ocean heat content is less sensitive to these choices. Although, choices in pulse size, background concentration, and model lead to uncertainties, the most important and subjective choice to determine AGWP of CO2 and GWP is the time horizon.
 
Last edited by a moderator:
More research:

http://www.climatechange2013.org/images/report/WG1AR5_Ch08SM_FINAL.pdf

So the AR5 calculation for $$\textrm{GWP}_{CH_4}(T) = \frac{\left(1 + 0.50 + 0.15 \right) \times \frac{28.97}{16.04} \times \frac{10^9}{5.1352\times10^{18}} \times 3.63 \times 10^{-4} \times 12.4 \times \left( 1 - e^{- \frac{T}{12.4}} \right)}{1.7517 \times 10^{-15} \times \left( 0.2173 \times T + 0.2240 \times 394.4\times \left( 1 - e^{- \frac{T}{394.4}} \right) + 0.2824 \times 36.54\times \left( 1 - e^{- \frac{T}{36.54}} \right)+ 0.2763 \times 4.304\times \left( 1 - e^{- \frac{T}{4.304}} \right) \right)}$$

Thus $$\textrm{GWP}_{CH_4}(0) = 120, \; \textrm{GWP}_{CH_4}(10) = 104, \; \textrm{GWP}_{CH_4}(20) = 83.8 , \; \textrm{GWP}_{CH_4}(50) = 48.4, \; \textrm{GWP}_{CH_4}(100) = 28.5-, \; \textrm{GWP}_{CH_4}(500) = 8.1$$ which agrees with the table in the two tabulated places.

The calculations for $$N_2O$$ are tedious, but the rest of the them are straightforward:

$$\textrm{GWP}_{CFC-11}(T) = \frac{\frac{28.97}{137.37} \times \frac{10^9}{5.1352\times10^{18}} \times 0.26 \times 45 \times \left( 1 - e^{- \frac{T}{45}} \right)}{1.7517 \times 10^{-15} \times \left( 0.2173 \times T + 0.2240 \times 394.4\times \left( 1 - e^{- \frac{T}{394.4}} \right) + 0.2824 \times 36.54\times \left( 1 - e^{- \frac{T}{36.54}} \right)+ 0.2763 \times 4.304\times \left( 1 - e^{- \frac{T}{4.304}} \right) \right)}$$
 
rpenner said:
... Thus $$\textrm{GWP}_{CH_4}(0) = 120, \; \textrm{GWP}_{CH_4}(10) = 104, \; \textrm{GWP}_{CH_4}(20) = 83.8 , \; \textrm{GWP}_{CH_4}(50) = 48.4, \; \textrm{GWP}_{CH_4}(100) = 28.5-, \; \textrm{GWP}_{CH_4}(500) = 8.1$$ which agrees with the table in the two tabulated places. ...
Click to expand...
Many thanks again. I credit you with finding that great link you first gave in my post commenting on parts of it here: http://www.sciforums.com/showthread.php?97892-Climate-gate&p=3197486&viewfull=1#post3197486

If my simple ideas are basically correct, I can invert your accurate results to say, for example, that that the Global Warming effect in 10 years of 100 tones of CH4 released at t = 0 now is the same as only 100(104/120) or only 86 & 2/3 tons released at t= 0 when there is no OH- (or other oxidizing agent like ozone) then in the air. This gives sort of a limit on how much stronger effect CH4 will become (self amplify) when earlier fluxes of it have removed most of the oxidation agents.
 
As far as longer-lasting carbon sinks go, many economists argue that the forestry industry, when conducted in a sustainable fashion, actually helps reduce greenhouse gas levels by removing vast amounts of carbon from the forests rather than waiting for them to go up in a natural fire, and making room for new growth to continue absorbing CO[sub]2[/sub].
 
CptBork said:
As far as longer-lasting carbon sinks go, many economists argue that the forestry industry, when conducted in a sustainable fashion, actually helps reduce greenhouse gas levels by removing vast amounts of carbon from the forests rather than waiting for them to go up in a natural fire, and making room for new growth to continue absorbing CO[sub]2[/sub].
Click to expand...
Yes, wooden furniture and homes can store carbon removed from the air for decades, provided dozens of acres of forest are not burned to hide the illegal selective cutting of a mahogany tree, etc. as is done in Brazil. The beneficial effect when done properly (replanting) would be even greater it cellulosic alcohol becomes feasible - run you car on waste sawdust and old newspapers, etc., as that displaces fossil fuel.
 
You must log in or register to reply here.
Back
Top